J. Embryol. exp. Morph. 90, 363-377 (1985) 363

Printed in Great Britain © The Company of Biologists Limited 1985

Measurement of biological shape: a general methodapplied to mouse vertebrae

D. R. JOHNSON, P. O'HIGGINS, T. J. McANDREW,L. M. ADAMSDepartment of Anatomy, Medical School, University of Leeds, Leeds LS2 9JT,U.K.

AND R. M. FLINNCentre for Computer Studies, University of Birmingham, U.K.

SUMMARY

A method is described for recording and analysing the projected shape of mouse vertebrae.The image of the shape is captured by a television camera, cleaned, digitized and subjected tomathematical analysis. A visual representation is obtained by reconstructing the shape in polarcoordinates about its centre of area. Further statistical analysis of the whole shape is performedafter a Fourier transform. This allows the shape to be represented by and reconstructed from 15numbers. The method does not rely on homologous points or expert opinion and allows meanshapes to be constructed. It successfully distinguished between 92 % of the test data, Tl and T2vertebrae from two strains of mice.

INTRODUCTION

A common problem facing the morphologist is the comparison of the shapes ofcomplex biological structures in a manner that allows full account to be taken ofnatural variation. The traditional answer to this problem has been to derive simplequantitative data which are suitable for univariate or multivariate statistical analy-sis. Traditionally these data have been in the form of linear and angular measure-ments taken between defined homologous points, or ratios of such measurements.In addition to the problems inherent in defining homologous points, this approachsuffers from the additional disadvantages that It ignores the frequently largeintervening regions and that it produces measurements which may be disconnectedfrom each other. In consequence so much information is lost that the originalshape, or even an approximation to it, cannot be reconstructed from the data.

This paper describes a system which retains as much of the information presentin a shape as possible for mathematical analysis and does not depend onhomologous points. The description falls into two parts, first the process of imagecapture and storage and secondly a discussion of the possible methods of analysisof the data so produced.

Key words: shape, mouse, vertebrae, computer, Fourier transform.

364 D. R. JOHNSON AND OTHERS

y

\

Fig. 1. The digitizing apparatus. A mouse second thoracic vertebra (T2) is placed onthe stage of the dissecting microscope (centre) fitted with a black and white televisioncamera. The image of the bone appears on the monitor (left). A digital image of theshape of the bone has been generated by the microcomputer and interface and isdisplayed on the computer screen (right).

MATERIALS AND METHODS

1. Image capture, processing and storageOur first attempts to capture outlines of bones relied on a digitizing pad. The outline of a

shape produced via a camera lucida or from a photographic print was digitized by tracing aroundit with a cursor. The accurate tracing of an outline proved to be excessively slow and laboriouswhile the transfer of an outline to a tracing and thence to a computer multiplied errors.

We found that the process of image capture could be speeded appreciably by the use of asimple video camera interface (VCI, Educational Electronics, 30 Lake St, Leighton Buzzard,Beds LU7 8RX, England). This low-cost unit (less than £200) allows a standard video signal tobe digitized by a microcomputer.

As examples of biological shapes we have chosen the anteroposterior projections of the firstand second thoracic vertebrae (Tl and T2 respectively) of two strains of mice: (1) the multiplerecessive strain (REC) is homozygous for the genes short ear (se), vestigial tail (vt), non-agouti(a), brown (b), dilute (d), pink eye (p), chinchilla (cc/l) and waved-2 (wa-2); (2) the F] (DOM) oftwo inbred strains C57BL and C3H which carries dominant alleles at all these loci. The papain-digested skeletons used are part of the material of Griineberg & McLaren (1972) and wereloaned by the British Museum (Natural History).

The mouse vertebra is placed on the illuminated base of a Wild M5a dissecting microscope(Fig. 1) and back lit. The microscope carries a standard C mount and is equipped with a blackand white video camera. The image of the bone, suitably magnified to fill the screen andreversed black/white (for the convenience of the operator) appears on a monitor screen and isalso fed to the VCI, which is in turn connected to a BBC (Acorn) microcomputer.

Once the synchronization and gain controls of the interface have been set to match the cameraan image can be captured and displayed on the computer screen (Fig. 2). This is achieved by

Measurement of shape 365

repeated sampling of the video signal and takes about four seconds. The BBC microcomputerworks in various display modes. The software supplied with the VCI allows an image consistingof 160x256 pixels (in 4 shades) to be digitized.

Once the image has been captured the standard software allows it to be dumped to printer orto disc. We have added further programs which allow additional manipulations:

(i) Clean up. This program converts the four shades of the original image to two (i.e. producesa black and white image) and sharpens the edge by an averaging method similar to that used inmany computer-enhancement programs. The screen memory is searched for a shade change andwhen one is detected the new shade is compared with that of its immediate neighbours and resetto conform to the majority. As well as cleaning the outline this program (which takes 45 sees)removes the image of dust from the background. We found accidentally that the image of ahuman hair laid across the vertebra was also removed.

(ii) Digitizing. This program finds the edge of a cleaned up image by sampling the screendiagonally. A vertebra is made up of two outlines, an outer representing the edge of the boneand an inner representing the border of the neural canal. The program allows the two outlines tobe digitized automatically from one object by default. A 'wandering probe' starts at the bottomleft corner of the screen and runs diagonally sampling locations 1,1; 2,2; 3,3 etc. until it locates ashade change at the edge of the shape. It then follows the edge of the outline until it has returnedto the start point. A second iteration of this program starts at the centre of the screen by default(but can be preset anywhere). If the preset is within a foramen the outline of the latter is foundand digitized as before. The outer outline is digitized clockwise and the inner anticlockwise toaid later identification (Fig. 3). More foramina can be identified by further defaults, or the'wandering probe' which is visible on screen can be set by means of a joystick (50 sees for twooutlines).

The data are stored on a disc as pairs of Cartesian coordinates then transferred to themainframe computer for further analysis.

Fig. 2. The reconstructed image of the vertebra captured by the apparatus as itappears on the computer screen.

366 D. R. JOHNSON AND OTHERS

IISST2 IIS iMir 123S Mtr Hilts. NESS 'W FM KM N 'f FN m i

• •• •• v

Fig. 3. The inner and outer outlines of the vertebra after digitization. The location ofeach spot on the screen is stored as a pair of Cartesian coordinates in a fixed order on acomputer disc.

2. Mainframe manipulation of the dataSuperimposition of outlines and fittingIn order to achieve a mean outline from a series of individual outlines the latter must be

superimposed. We do this in three stages. The outlines are first scaled to a standard area. Thecentre of area (centroid) of each is then found by integration and the shape re-expressed as 128polar coordinates centred on this point (128 is a perfect square and a perfect square is needed forthe fast Fourier transform, see below). This process allows superimposition of outlines upontheir centroids.

Because the bones have been placed in no special orientation upon the microscope stage it isalso necessary to rotate them relative to each other. In practice this is done by making a leastsquares fit comparison for each outline against a standard. The outline is rotated by one polarcoordinate at a time and the fit, relative to the standard, repeated. After complete rotation oneorientation will be found to give a minimum value for the sum of the differences of the squareson each polar radius; this is designated best fit. The method used is a modification of the leastsquares fit described by Sneath (1967). Sneath took as his measurement of fit the size of the sumof the squared distances between a series of 'homologous landmarks' expressed in Cartesiancoordinates. We have used as a measure of fit the residual area when shapes are superimposedupon their centroids. This is a function of the sum of the differences on each individual radiussquared:

r-iN

Residual area oc > V(rn2-Rn

2)2

where rn, Rn are corresponding polar coordinates on two shapes, N = number of polarcoordinates used and n is large.

Measurement of shape 367

Standard shapes were chosen by trial and error according to symmetry. A circle is obviouslyinappropriate because the residual area is equal in all orientations, but a semicircle is apossibility. In practice we used a simple polygon derived from a vertebral outline. Fig. 4illustrates the process of fitting. The reference axis was taken as the midline of the standardshape and the start point for Fourier analysis (see below) as the point at which the ventral side ofthe outer outline of the vertebra crosses this axis.

ErrorsErrors in the procedure described above may arise from two sources, videodigitization and

manipulation of data. To ascertain the size of these errors a test of technique was performed.This consisted of sequentially capturing 20 images of the same bone which was removed from themicroscope stage and replaced in a different position and orientation after each capture. The 20sets of Cartesian coordinates so generated were passed to the Amdal and fitted to a standardshape as described above.

Fig. 4. Deriving the best fit. (Top) The image of the vertebra, reconstituted from thecoordinate stream and re-expressed in polar coordinates about its centre of area lies inno particular orientation. Upon it is superimposed a standard shape of equal areaorientated about X and Y axes. (Bottom) After a least squares fit has been performedthe image of the vertebra is rotated to the position of best fit.

368 D. R. JOHNSON AND OTHERS

Further manipulationOnce all outlines in a group have been orientated with respect to a standard in this way the

mean value of each polar coordinate is calculated and a mean outline generated (Fig. 5).

Comparison of group meansGroup mean outlines can now be plotted on top of each other. The significance of the

difference at any polar radius can be estimated by a t test, or an estimate of total shape similaritycan be made.

Fitting an unknown outline to group meansA single individual of unknown provenance can be assessed for fit to any number of group

means. A bone having a good fit (low total sum of differences of squares) with one group and ahigh total sum of differences of squares with another is likely to be a member of the first group.

Fourier transformsThe mathematical techniques applied to the data so far have been extremely simple. The

opened out graph of polar coordinates can, however, be regarded as a waveform and sophisti-cated techniques already developed for waveform analysis in physical sciences and telecom-munications applied. Jean Baptiste Fourier (1768-1830) described a method which splits acomplex waveform into a series of sine and cosine components of varying amplitude. Lestrel(1974) applied the series to biological shapes. The general Fourier series can be represented as:

F(0) = a0 + ax cos 6 + bi sin 6 + SL2 COS20 + b2 sin20... an cosnd + bn sin nd

where ao is a constant, ai-an are known as cosine components, bi-bn are known as sinecomponents, and F(0) is the magnitude of a polar radius r.

Because the sine and cosine components are 90° out of phase the Fourier series can describehighly irregular waveforms by means of a series of numbers (Lestrel, 1982). An alternativenotation uses amplitude and phase lag instead of sine and cosine components.

R n cos (n0+ n)[n=l

where Ro-Rn are known as amplitude components and 0i-0n are known as phase lagcomponents. The amplitude/phase lag notation has the advantage over sine/cosine notationthat the amplitude coefficients are independent of the start point of the waveform. It does not,however, allow shape reconstruction.

A simple shape is adequately described by the early harmonics of the Fourier series: a morecomplex one will require more components to describe it accurately. A corollary of this is thatearly components of the series describe gross features of the shape and later ones fine detail. Inpractice the 'fine detail' may represent noise in the measurement process and may often bediscarded without detriment to shape analysis. Since we are dealing with several pairs ofcomponents, a multivariate statistical approach, which considers all variates simultaneously, isappropriate. We used the DISCRIM procedure within SAS (Statistical Analysis System; SASusers guide 1982) which calculates the generalized squared distances between each test indi-vidual and the calibration groups, and classifies them on the basis of 'nearest group'.

The first pair of Fourier coefficients (a0, b^ are excluded because the first cosine component isconstant (since areas are equalized, Lestrel, 1974) and the first sine component is always zero.The first fifteen pairs of coefficients referred to hereafter are therefore coefficient numbers 2-31inclusive.

RESULTS

Errors

Capture

The unit of resolution of the television screen is the illuminated dot, the pixel. Ifthe image of a bone occupies any part of a pixel the latter will be illuminated. This

Measurement of shape 369

is obviously the source of a small error. With the magnification used (x50objective plus xO-3 correcting lens on the microscope, 16" colour monitor) theimage of a test T2 vertebra had a maximum height of 152 mm and width of172mm 1 pixel measured 0-5 mm highxl-Omm wide. The maximum error due tothis source was thus less than 1 %.

Computer rounding error

The test of technique on a single vertebra allows us to measure the sum ofcapture and computing errors. The 128 mean values produced by the test oftechnique had a mean variance ratio

/standard deviation\ mean

Vertebral comparisons

Fig. 5 shows the computed mean outer outlines for 22 Tls and 14 T2s from thedominant strain. The computer-generated plot gives a polar reconstruction ofvertebral shape (left) and an opened out linear plot (right). The trace representsthe mean outline ± 10 standard errors of the mean. The more usual representationof mean ± 2 standard errors plots as a single line.

Fig. 6 shows comparison plots of Tls (upper) and T2s (lower) from dominantand recessive strains. Significant differences (P< 0-05) are present in many areas.

Table 1 shows the results of fitting individual Tls to group mean shapes. 44 outof 53 bones (83 %) fitted best to their 'correct1 group. A similar comparisonamongst the T2s gave 32 out of 34 (94 %). Overall 87 % were correctly classified.

The Fourier series will adequately describe a shape in less than 128 variables(the number of polar coordinates chosen) and so potentially simplifies the data.Fig. 7 shows reconstructions of a T2 vertebra based on 5-60 sine/cosine coefficientpairs. It can be seen that 15 pairs subjectively appear to describe the shapeadequately and further coefficients add little to definition.

Univariate analysis of the first 15 coefficient pairs was undertaken. Examples ofbar charts showing the upper and lower 95 % confidence limits and mean ± 2 S.E.M.for each population (group) are reproduced in Fig. 8. All cosine componentsshowed significant differences between group means at the level of P< 0-0001(Table 2), some discriminating between Tl and T2 and some between DOM andREC. Only 4 of 15 sine components were significant at this level. No singlecoefficient split all 4 groups unequivocally.

An objective test of the decision to analyse only 15 pairs of coefficients is toperform a discriminant function analysis which compares all variates simul-taneously. For this a random sample of 10 % of all vertebrae was removed fromthe data set and an attempt made to classify them with respect to the remainder.This was repeated 10 times using 5,10,13,15,18 and 20 coefficient pairs. The bestclassification of this data set (92 % correct) was obtained using 15 sine/cosine pairs(Fig. 9). If fewer or more pairs are used definition suffers; below 15 pairs the

370 D. R. JOHNSON AND OTHERS

50 100 150 200 250Angle (degrees)

300 350

Fig. 5. Mean vertebral shapes for 21 first (Tl) and 14 second (T2) thoracic vertebraefrom the DOM strain of mice. Thick line, mean: thin lines, mean ± IOXS.E.M.

100 150 200 250Angle (degrees)

300 350

Fig. 6. Comparison plots of means of 22 DOM Tls and 14 T2s (thick lines)superimposed upon 31 REC Tls and 20 T2s (thin lines). The horizontal bars show theareas where the shapes differ significantly at the level P< 0-05.

shapes are poorly defined, above 15 pairs the effects of sample size and noiseintrude. 79 out of 87 (91 %) of shapes were correctly classified using 15 cosinecomponents only.

Measurement of shape 371

Table 1. Fits of individual Tl vertebrae against group meansBone

number

DM95 TlDM97 TlDM98 TlDM99 TlDM100 TlDM101 TlDM102 TlDM103 TlDM104 TlDM106 TlDM107 TlDM108 TlDM109 TlDM110 TlDM111 TlDM112 TlDM113 TlDM114 TlDM115 TlDM116 TlDM117 TlDM119 TlREC20T1REC23T1REC25T1REC34T1REC35T1REC36T1REC37T1REC38T1REC39T1REC40T1REC41 TlREC42T1REC43T1REC44T1REC45T1REC46T1REC47T1REC49T1REC50T1REC51T1REC52T1REC53T1REC54T1REC55T1REC57T1REC58T1REC59T1REC60T1REC62T1REC63T1REC65T1

Fit to recessive mean(total sum of squares)

10092640195923071931667

2609134412321514908

19262251773

21871975168715082611463222241251702377738643

13388653529600

1317778348876

24802480924

1021877568

15931721846

1239689

2744153416021304880

1478664854

* Denotes misclassification.

Fit to dominant mean(total sum of squares)

911855848826

1520923

1089611519706522

1000813

1389572962953698

11992493855

11291950996

1203115415367056475

22523477819

1439247953405340254318781248183536441522156611741706598811231231118823003703724

1683

Decision

DDDDDR*DDDDDDDR*DDDDDDDDRRRRRD*D*RRRRRRRRRRRRD*RD*RRD*D*D*RRRR

372 D. R. JOHNSON AND OTHERS

DISCUSSION

The final shape attained by a bone must be dependent upon a host of factors,both genetical and environmental. The almost universal occurrence of pleiotropy(multiple effects of genes on characters) has led to the hypothesis that totalphenotype is acted upon by selection and that it is this which evolves rather thanindividual characters or genes (Wright, 1968; Cheverud, 1982). Integrated systemsare now emphasized in morphogenesis (Waddington, 1957; Leamy, 1977; Riedl,1978; Lande, 1979; Atchley, Rutledge & Cowley, 1981; Cheverud, 1982; Bonner,1982). If we view a bone as an integrated system then we must ask how best tomeasure its total shape.

In the conventional methods of comparing bone shapes homologous points aredefined in such a manner as to permit measurements which reflect individualfeatures thought to be of biological significance and which can be taken quicklyand consistently. In practice we suspect that the latter consideration often out-weighs the former. Thus Festing (1972) chose 13 measurements of the mousemandible which could be read off 'as quickly as they could be recorded by anassistant', Atchley (1983) used eight traits 'chosen because they are easilymeasured and the measurements are highly repeatable' (rat mandible) and Leamy& Atchley (1983) used 19 scapular measurements 'taken from well definedlandmarks to optimise repeatability'. Multivariate analysis will remove corre-lations between such measurements so that they are mathematically respectable,but their biological significance must remain in doubt.

The technique described here does not rely upon homologous points. No startpoint for the coordinate stream is specified: the only defined point is the centre ofarea, which is a relatively neutral property of the shape. The number of polarcoordinates generated depends upon the video system used. More than 128 points

20

Fig. 7. A mouse T2 vertebra (REC60 T2) reconstructed from 5-60 pairs of sine/cosinecoefficients.

0-68

Measurement of shape

2-21 -5-03

017

-108-8

—

DOMT2 -

RECT2

- DOMT1

- RECT1

- r -

Cos 2

2-72

Cos 3

9-77

Cos 4

-20-6 19-7

-6-96 14-63

Fig. 8. Bar charts showing mean ± 2 S.E.M. (this line) and 95 % confidence limits (thinline) of the first three pairs of cosine and sine coefficients of the Fourier series. Notethat the cosine components are better discriminators than the sines.

could, of course, be generated from a video system giving higher resolution. Theseare easily obtainable, but expensive.

Because the system generates a reconstructed shape rather than a series ofnumbers we need to think about analysis of results in a different way, derivingfunctions which relate to the shape as a whole rather than arbitrary measurementswithin it. Our simple polar plot superimposition allows variation to be taken intoaccount, provides acceptable discrimination betv/een shapes and tells us whetherthe difference is significant at a particular point or in a particular area. Using polarcoordinates homologous points (the tips of the transverse processes, for example)will not necessarily map on the same radius in two compared shapes. The nth polar

374 D. R. JOHNSON AND OTHERS

Table 2. Variance ratio (F) values for the first fifteen pairs of Fourier coefficients andassociated probabilities (T)

Pair

2345678910111213141516

cosine

F291-971224-70378-9259-88

273-11365-94135-9769-02

269-9352-7766-41101-9762-858-08

72-01

P< 0-0001< 0-0001< 0-0001< 0-0001< 00001< 0-0001< 0-0001< 0-0001< 0-0001< 00001< 0-0001< 00001< 00001< 0-0001< 0-0001

F

6-031-280-852-338-407-160-985-8511-815-342-85141913-772-612-85

sine

P< 0-0010< 0-2780< 0-4750< 00794< 0-0001< 0-0003< 0-4066< 0-0012< 0-0001< 00022< 0-0414< 0-0001< 0-0001< 0-0588< 0-0415

radius, encompassing the tip of the transverse process in shape A, should not,therefore, be compared blindly with the nth polar radius in shape B which missesit. If the nth radii between two shapes differ, then the shapes differ.

It can be seen from the data of Table 1 that a good fit to the dominant strain doesnot necessarily indicate a poor fit to the recessive shape and vice versa. This isbecause the shapes are highly irregular and a bone fitting the dominant shape wellin some areas may fit a recessive shape well in others. Because of this furtherstatistical analysis (such as probability of group membership) using this routinewas not attempted. We also suspect that the populations of shapes may overlap insome cases, so that some individuals could be a member of either population.Moore & Mintz (1972) however were able to identify coded bones from C3H andC57BL mice with 85-100% accuracy. The variation in inbred strains would, ofcourse, be lower than in our material.

Each Fourier component, unlike each polar radius, represents a property of thewhole outline. The problem of homologies is thus minimized in that each Fouriercomponent is dependent only on the centroid (and in the case of sine/cosinecomponents on the start point which is derived from our fitting routine, notarbitrarily specified by eye). Fifteen Fourier coefficient pairs can classify the shapeas well as 128 polar coordinates and fifteen cosine components as well as fifteensine/cosine pairs. The number 15 is probably a function of the particular shapesused and the variance within the data set: other sets of data describing bones ofdifferent shapes might well need more or fewer components to best describe them.

Since the Fourier procedure is a simple transformation of the original data itsdiscrimination cannot exceed that of the former. Less than 15 pairs of componentswill simplify the shape (Fig. 7) and thus inhibit discrimination. The use of morethan 15 component pairs cannot increase the discrimination further, but should

Measurement of shape 375

not diminish it. In fact more pairs reduce discrimination a little: we suggest thatthis is due to noise, i.e. cumulative errors in the system.

It is a property of the Fourier series that sine components describe axialasymmetry (Zahn & Roskies, 1972; Lestrel, 1974): since the vertebral shapes areessentially symmetrical about their midline they can be reconstructed minus anyasymmetry from the first 15 cosine components alone (Fig. 10, cf. Fig. 7).

Fourier components as used in this context are based upon a centroid and thusreflect the disposition of the shape about this point relative to the start point. Useof amplitude coefficients only would remove the start point dependency, but notthat on the centroid. Because we are dealing with similar shapes the effects ofcentroid dependency are minimized and differences in Fourier components reflectdifferences in shape. The Fourier analysis of an edge-based decomposition of ashape (e.g. the tangent/angle function, Bookstein, 1977; Zahn & Roskies, 1972)would remove centroid dependency also.

We suggest that an ideal system for shape measurement should conform to thefollowing criteria:

1. It should be practical and practicable.2. It should accurately measure the form or any part of it.3. It should allow reconstruction of the original shape (i.e. the derived

measures should be related to the shape by a determinable function).4. The data should be suitable for statistical analysis, so that biological

variation can be accommodated.5. Measurements of size should be independent of shape and vice versa.6. Measurements of shape should, if required, be independent of any necessity

to define 'homologous points'.

10-

10—i 1 1 r

15 20

No. of variate pain;

Fig. 9. Regression curve of percentage of bones misclassified against number ofcoefficient pairs used.

376 D . R. JOHNSON AND OTHERS

20

Fig. 10. A mouse T2 vertebra (REC60T2) reconstructed from 5-20 Fourier cosinecoefficients.

The Fourier method described here conforms to the first five of these desider-ata: Fourier analysis of a curvature function would conform to all six.

Any new method of measuring shape must offer significant advantages overexisting methods. The system described here is of the same order of accuracy asexisting methods and, we think, offers considerable advantages. For furthercomparison of traditional and more modern methods of analysis the reader isreferred to Ashton, Flinn, Moore & O'Higgins (in preparation).

The ultimate test of shape measurement must be its ability to 'recognize'unknown shapes. The Fourier method described here classified 92 % of the sampleof outlines correctly. 15 coefficients describe total shape with a high measure ofaccuracy, with no reliance upon expert opinion or 'homologous points'. Wesuggest that it should now be possible, using suitable material, to partition size andshape allowing the complex interrelations of these properties to be studied in abiological context.

REFERENCESASHTON, E. F., FLINN, R. M., MOORE, W. J. & O'HIGGINS, P. (1986). Cranial shape and hominoid

classification.ATCHLEY, W. R. (1983). A genetic analysis of the mandible and maxilla in the rat. /. Craniofac.

Genet, devl Biol. 3, 409-422.

Measurement of shape 377

ATCHLEY, W. R., RUTLEDGE, J. J. & COWLEY, D. E. (1981). Genetic components of size and shape.ii Multivariate covariance patterns in the rat and mouse skull. Evolution, Lawrence, Kans. 35,1037-1055.

BOOKSTEIN, F. L. (1977). The Measurement of Biological Shape and Shape Change. Ph.D.thesis, University of Michigan.

BONNER, J. T. (1982). Evolution and Development. Berlin: Springer Verlag.CHEVERUD, J. (1982). Phenotype, genetic and environmental morphological integration in the

cranium. Evolution, Lawrence, Kans. 36, 499-516.FESTING, M. (1972). Mouse strain identification. Nature, Lond. 238, 351-352.GRUNEBERG, H. & MCLAREN, A. (1972). The skeletal phenotype of some mouse chimeras. Proc.

R. Soc. Lond. B 182, 9-23.LANDE, R. (1979). Quantitative genetic analysis of multivariate evolution applied to brain: body

size allometry. Evolution, Lawrence, Kans. 33, 402-416.LEAMY, L. (1977). Genetic and environmental correlations of morphocentric traits in

randombred house mice. Evolution, Lawrence, Kans. 31, 357-369.LEAMY, L. & ATCHLEY, W. R. (1984). Morphometric integration in the rat (Rattus sp.) scapula.

/. Zool. (Lond.) 202, 43-56.LESTREL, P. E. (1974). Some problems in the assessment of morphological size and shape

differences. Yearb. Phys. Anthrp. 18, 140-162.LESTREL, P. E. (1982). A Fourier analytic procedure to describe complex morphological shapes.

In Factors and Mecanisms Influencing Bone Growth (ed. A. D. Dixon & B. G. Sarnat). NewYork: Alan R. Liss, Inc.

MOORE, W. J. & MINTZ, B. (1972). Clonal model of vertebral column and skull developmentderived from genetically mosaic skeletons in allophenic mice. Devi Biol. 27, 55-70.

RIEDL, R. (1978). Order in Living Organisms. New York: Wiley & Sons.SAS USERS GUIDE: STATISTICS (1982). Pp. 381-396. SAS Institute, Cary N.C.SNEATH, P. H. A. (1967). Trend-surface analysis of transformation grids. J. Zool. (Lond.) 151,

65-122.WADDINGTON, C. H. (1957). The Strategy of the Genes. London: Allen & Unwin.WRIGHT, S. (1968). Evolution and the Genetics of Populations 1. Genetic and Biomedical

Foundations. Chicago: University of Chicago Press.ZAHN, C. T. & ROSKIES, R. Z. (1972). Fourier descriptors for plane closed curves. IEEE Trans.

Comput. C-21, 269-281.

{Accepted 29 July 1985)